Solution%20to%20HW_4

# Solution%20to%20HW_4 - Solution to HW#4 EGM 6812 Fall 2009...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Solution to HW#4 EGM 6812 Fall 2009 1. Using the differential conservation of momentum equation as a starting point, derive the vorticity transport equation, ( 29 2 D V Dt ϖ ϖ ν ϖ = ⋅ ∇ + ∇ r r r r , for a flow with constant density and viscosity. Hints: start by taking 2 b DV f p V Dt ρ ρ μ ∇ × =- ∇ + ∇ r r r . Solution: (note both the underline and upper arrow denote vector) Note: b f ρ ∇ × = r assuming b f r is only due to gravitational force OR b f r is conservative p ∇ ×∇ ≡ 2 2 V μ μ ϖ ∇ ×∇ = ∇ ur r ( ) DV V V Dt t ϖ ρ ρ ∂ ∇ × = + ∇× ⋅∇ ∂ ur r r r Since 1 ( ) ( ) 2 V V V V V V ⋅∇ = ∇ ⋅- × ∇× r r r r r r => ( 29 ( 29 V V V ϖ ∇ × ⋅∇ =- ∇ × × ur r r r Since ∇ x (A x B ) = A ∇⋅ B- A ⋅∇ B- B ∇⋅ A + B ⋅∇ A , we have ( 29 V V V V V ϖ ϖ ϖ ϖ ϖ ∇ × × = ∇ ⋅- ⋅∇- ∇ ⋅ + ⋅∇ ur ur ur ur ur r r r r r Thus, the LHS of the equation becomes...
View Full Document

## This note was uploaded on 09/09/2010 for the course EGM 6812 taught by Professor Renweimei during the Fall '09 term at University of Florida.

### Page1 / 3

Solution%20to%20HW_4 - Solution to HW#4 EGM 6812 Fall 2009...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online